Genesis 6:15 in the Bible tells us the Ark's dimensions were at least 135 meters long (300 cubits), 22.5 meters wide (50 cubits), and 13.5 meters high (30 cubits). That's 450 feet long, 75 feet wide, and 45 feet high! It could have been larger, because several larger-sized cubits were used. But the 45-centimeter (18-inch) cubit is long enough to show the enormous size of the Ark. It is typically measured from the tip of the finger to the elbow.
According to this article there are 1,250,000 species of animal. Noah brought on board only those animals with the "breath of life" in their nostrils. Eliminate the invertebrates and fish and insects. I am now coming up with a loose number of 30,500 animals that needed to be saved according to today's standards.
But another point to consider is the variety of species. Have you ever heard of a Puggle (pug + beagle)? A Labra-doodle? (lab + poodle). These are relatively new breeds of dogs that did not exist thousands of years ago. Rather than a wolf, coyote, fox, dog, there was probably one common ancestor. This does not prove evolution, but rather the principles of eugenics.
The same can be said for wild animals that breed with each other (lion + tiger) or even beasts of burden such as a mule (horse + donkey). My point is: there were probably not as many species 4,500 years ago before the flood. There was probably a common ancestor for the animals we have today, but not as many as there are today. Based on the dimensions of the ark, and the fact that Noah probably was smart enough to take baby animals on board (because they would save space) it is totally plausible.
Belief has nothing to do with it. It is explained with math and science. The problem is whether folks are willing to put the time in to figuring it out. The Bible says to meditate on the scriptures daily. Meditation is not emptying your mind as pagans would have you do. Meditation in the biblical sense is to study repeatedly and focus on one word or phrase.
Personally, I believe that a cubit would have been larger than 18 inches. We find evidence of large animals from the past (giant sloth, dinosaurs, giant turtles, sharks, etc.) There is no reason to believe that humans would not have been larger, too. The Bible teaches that there was lush vegetation over the entire earth with some lakes and small seas. There was also a protective layer of atmosphere made of ice or water. This would filter out radiation, compress the atmosphere to increase air pressure and act as a thermal blanket. More detail is needed, but I'm on my lunch break now -- but these conditions would stand up to scientific scrutiny, and would explain how things would grow larger and live longer. Therefore I believe that the Ark was probably larger than described above if one uses the fingertip to elbow measurement for a cubit.